Parallelogram Side Length: Finding AC When Perimeter = 21 and AB = 7

Q: Why don't we add all four sides of the parallelogram?

Great question! In a parallelogram, opposite sides are equal. So if AB = 7, then CD = 7 too. If AC = 0.5X, then BD = 0.5X. The formula P = 2(side₁ + side₂) automatically accounts for both pairs!

Q: How do I know which sides are adjacent in the parallelogram?

Look at the diagram! Adjacent sides share a common vertex (corner). In this parallelogram, AB and AC share vertex A, so they're adjacent. Use these two different side lengths in your formula.

Q: What if I get a different value for X?

Double-check your algebra! Start with $ 21 = 2(7 + 0.5X) $, then divide by 2: $ 10.5 = 7 + 0.5X $. Subtract 7: $ 3.5 = 0.5X $. Finally, divide by 0.5: $ X = 7 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find AC

00:03 Opposite sides are equal in parallelograms

00:10 They are also a pair of opposite sides therefore equal

00:21 The perimeter of the parallelogram equals the sum of its sides

00:35 Let's substitute appropriate values and solve for X

00:51 Let's isolate X

00:58 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Shown below is a parallelogram.

AB = 7

AC = 0.5X

The perimeter of the parallelogram is 21.

Calculate side AC.

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the formula for the perimeter of a parallelogram.
Step 2: Substitute the given values into the perimeter formula.
Step 3: Solve for the unknown variable $X$ .
Step 4: Determine the specific length of AC using the value of $X$ .

Now, let's work through each step:

Step 1: The formula for the perimeter of a parallelogram is given by

P = 2(a + b)

where $a$ and $b$ are the lengths of the two pairs of sides.

Step 2: Substitute the known values into the formula. Here, let $a = AB = 7$ and $b = AC = 0.5X$ .

Perimeter $= 21$ , so:

21 = 2(7 + 0.5X)

Step 3: Solve for the unknown variable $X$ .

First, divide both sides of the equation by 2 to isolate the terms inside the parenthesis:

10.5 = 7 + 0.5X

Subtract 7 from both sides:

3.5 = 0.5X

Multiply both sides by 2 to isolate $X$ :

X = 7

Step 4: Determine the length of AC:

Substitute back to find $0.5X = AC$ :

AC = 0.5 \times 7 = 3.5

Therefore, the length of side AC is $3.5$ .

3

Final Answer

3.5

Key Points to Remember

Essential concepts to master this topic

Perimeter Formula: P = 2(side₁ + side₂) for parallelograms
Substitution: 21 = 2(7 + 0.5X) leads to X = 7
Verification: AC = 0.5(7) = 3.5, so perimeter = 2(7 + 3.5) = 21 ✓

Common Mistakes

Avoid these frequent errors

Using all four sides instead of two pairs
Don't add AB + BC + CD + DA = 7 + 0.5X + 7 + 0.5X! This creates the wrong equation 14 + X = 21, giving X = 7 and AC = 3.5, which seems right but uses flawed logic. Always remember parallelograms have opposite sides equal, so use P = 2(adjacent sides).

Practice Quiz

Test your knowledge with interactive questions

Given the parallelogram:

Calculate the perimeter of the parallelogram.

FAQ

Everything you need to know about this question

Why don't we add all four sides of the parallelogram?

+

Great question! In a parallelogram, opposite sides are equal. So if AB = 7, then CD = 7 too. If AC = 0.5X, then BD = 0.5X. The formula P = 2(side₁ + side₂) automatically accounts for both pairs!

How do I know which sides are adjacent in the parallelogram?

+

Look at the diagram! Adjacent sides share a common vertex (corner). In this parallelogram, AB and AC share vertex A, so they're adjacent. Use these two different side lengths in your formula.

What if I get a different value for X?

+

Double-check your algebra! Start with $21 = 2(7 + 0.5X)$ , then divide by 2: $10.5 = 7 + 0.5X$ . Subtract 7: $3.5 = 0.5X$ . Finally, divide by 0.5: $X = 7$ .

Can AC ever be longer than AB in this problem?

+

Let's see! Since AC = 0.5X and we found X = 7, we get AC = 3.5. Since AB = 7, we have AB > AC. The coefficient 0.5 ensures AC will always be half of X, making it smaller when X equals AB.

How do I check if my final answer is correct?

+

Substitute back into the perimeter! If AC = 3.5 and AB = 7, then perimeter = 2(7 + 3.5) = 2(10.5) = 21. This matches the given perimeter, so AC = 3.5 is correct! ✓

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